Three-dimensional fundamental solutions for transient heat transfer by
conduction in an unbounded medium, half-space, slab and layered media
Antonio Tadeu *, Nuno Simoes
Faculty of Sciences and Technology, Department of Civil Engineering, University of Coimbra, Pinhal de Marrocos, Polo II, 3030-290 Coimbra, Portugal
Received 11 August 2005; accepted 24 January 2006
Available online 5 April 2006
Abstract
This paper presents analytical Green’s functions for the transient heat transfer phenomena by conduction, for an unbounded medium, half-
space, slab and layered formation when subjected to a point heat source. The transient heat responses generated by a spherical heat source
are computed as Bessel integrals, following the transformations proposed by Sommerfeld [Sommerfeld A. Mechanics of deformable bodies.
New York: Academic Press; 1950; Ewing WM, Jardetzky WS, Press F. Elastic waves in layered media. New York: McGraw-Hill; 1957].
The integrals can be modelled as discrete summations, assuming a set of sources equally spaced along the vertical direction. The expressions
presented here allow the heat field inside a layered formation to be computed without fully discretizing the interior domain or boundary interfaces.
The final Green’s functions describe the conduction phenomenon throughout the domain, for a half-space and a slab. They can be expressed as
the sum of the heat source and the surface terms. The surface terms need to satisfy the boundary conditions at the surfaces, which can be of two
types: null normal fluxes or null temperatures. The Green’s functions for a layered formation are obtained by adding the heat source terms and a set
of surface terms, generated within each solid layer and at each interface. These surface terms are defined so as to guarantee the required boundary
conditions, which are: continuity of temperatures and normal heat fluxes between layers.
This formulation is verified by comparing the frequency responses obtained from the proposed approach with those where a double-space
Fourier transformation along the horizontal directions [Tadeu A, Antonio J, Simoes N. 2.5D Green’s functions in the frequency domain for heat
conduction problems in unbounded, half-space, slab and layered media. CMES: Computer Model Eng Sci 2004;6(1):43–58] is used. In addition,
time domain solutions were compared with the analytical solutions that are known for the case of an unbounded medium, a half-space and a slab.
q 2006 Elsevier Ltd. All rights reserved.
Keywords: Transient heat transfer; Conduction; 3D Green’s functions; Layered media
1. Introduction
This work presents Green’s functions for calculating the
three-dimensional transient heat transfer by conduction in the
presence of an unbounded, half-space, slab and multi-layer
formations when heated by a point source. The problem is
formulated in the frequency domain using time Fourier
transforms. The proposed technique allows the use of any
amplitude time evolution of the heat source. The proposed
fundamental solutions relate the heat field variables (fluxes or
temperatures) at some position in the domain caused by a heat
source placed elsewhere in the media.
This work extends the previous work carried out by the
authors to define the heat conduction response of layered solid
0955-7997/$ - see front matter q 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.enganabound.2006.01.011
* Corresponding author. Tel.: C351 239 797201; fax: C351 239 797190.
E-mail address: [email protected] (A. Tadeu).
media subjected to a spatially sinusoidal harmonic heat line
source [3]. The technique described in that paper requires the
knowledge of the Green’s functions for the unbounded media.
They are developed by first applying a time Fourier transform
to the time diffusion equation for a heat point source and then a
dual spatial Fourier transform to the resulting Helmholtz
equation, along the two horizontal directions (x and z), in the
frequency domain. So these functions are written first as a
superposition of cylindrical heat waves along one horizontal
direction (z) and then as a superposition of heat plane sources.
The Green’s functions for a layered formation were
formulated as the sum of the heat source terms equal to those
in the full-space and the surface terms required to satisfy the
boundary conditions at the interfaces, i.e. continuity of
temperatures and normal heat fluxes between layers, and
null normal fluxes or null temperatures at the outer surface.
The total heat field was found by adding the heat source terms,
equal to those in the unbounded space, to the sets of surface
terms arising within each layer and at each interface.
Engineering Analysis with Boundary Elements 30 (2006) 338–349
www.elsevier.com/locate/enganabound
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349 339
In the work described here, the heat diffusion generated by a
spherical heat source is computed as a Bessel integral,
following the transformations proposed by Sommerfeld [1,2]
in his study on the propagation of elastic waves. The resulting
integrals are made discrete, assuming the existence of a set of
sources equally spaced along the vertical direction.
In the presence of a layered formation, the heat diffusion
within each layer is modelled using, at the two layer interfaces,
surface terms that are similar to the source, with unknown
amplitudes a priori. The amplitudes of these surface terms are
defined by prescribing the existing boundary conditions at the
solid interfaces (i.e. continuity of temperatures and normal heat
fluxes between layers, and null normal fluxes or null
temperatures at the outer surfaces).
These Green’s functions are useful in themselves. However,
they can also be incorporated into boundary element method
(BEM) or method of fundamental solutions (MFS) algorithms
to model the heat diffusion across layered media containing
three-dimensional heterogeneities, without the discretization of
the flat horizontal interfaces. The present Green’s functions
require much less computational effort than the earlier ones.
However, it should be noted that the previously developed
Green’s functions, which require a dual spatial Fourier
transform, are more suitable for solving layered media
containing cylindrical heterogeneities.
This paper first presents the three-dimensional solution of a
point heat diffusion source in an unbounded medium,
explaining the mathematical manipulation required to obtain
the solution as a Bessel integral in the frequency domain. The
procedure to retrieve the time domain solutions is also given.
This methodology is verified by comparing the results obtained
with the exact time solutions.
This paper then goes on to describe the formulation for a
point heat load applied to a half-space, a slab, a slab over a half-
space medium and a layered formation. The continuity of
temperature and heat fluxes need to be established between two
neighbouring layers, while null temperatures or null heat fluxes
may be prescribed along an external interface of the boundary.
The full set of expressions is corroborated by comparing its
frequency solutions with those provided by the previous
formulation that requires a double spatial Fourier transform.
Time solutions are verified in the case of a half-space and a
slab, for which analytical solutions are known, by using an
image model approach (see Carslaw and Jaeger [4]).
2. 3D problem formulation and Green’s functions in an
unbounded medium
The transient heat transfer by conduction in the x, y and z
directions is expressed by the equation
v2
vx2C
v2
vy2C
v2
vz2
� �T Z
1
K
vT
vt; (1)
in which t is time, T(t,x,y,z) is temperature, KZk/(rc) is the
thermal diffusivity, k is the thermal conductivity, r is the
density and c is the specific heat. The application of a Fourier
transformation in the time domain to Eq. (1) gives the equation
v2
vx2C
v2
vy2C
v2
vz2
� �C
ffiffiffiffiffiffiffiffiffiKiu
K
r !2 !Tðu; x; y; zÞZ 0; (2)
where iZffiffiffiffiffiffiK1
pand u is the frequency. For a heat point
source, applied at (x0,y0,z0) in an unbounded medium, of the
form pðu; x; y; z; tÞZdðxKx0ÞdðyKy0ÞdðzKz0ÞeiðutÞ, where
d(xKx0), d(yKy0) and d(zKz0) are Dirac-delta functions, the
fundamental solution of Eq. (2) can be expressed as
T fðu; x; y; zÞZeKi
ffiffiffiffiffiffiffiffiffiffiffiKðiu=KÞ
pr0
2kr0
(3)
with r0ZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxKx0Þ
2C ðyKy0Þ2C ðzKz0Þ
2p
.
This solution can be expressed as a Bessel integral,
following the transformations proposed by Sommerfeld [1,2]
T fðu; x; y; zÞZKi
2k
ðN0
J0ðkyrÞky eKi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKðiu=KÞKk2
y
pjyKy0jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
KiuKKk2
y
q dky; (4)
where J0( ) are Bessel functions of order 0, ky is the
vertical wavenumber along the y direction and
rZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxKx0Þ
2C ðzKz0Þ2
p.
This integral can be expressed as a discrete sum if we
assume the existence of virtual sources, equally spaced at Ly,
along the vertical direction y,
T fðu; x; y; zÞZKip
Lyk
XNnZ1
J0ðknrÞknnn
eKinnjyKy0j (5)
with nnZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKðiu=KÞKk2
n
p, knZ(2p/Ly)n. The distance Ly is
chosen so as to prevent spatial contamination from the virtual
sources, i.e. it must not be too small [5].
2.1. Responses in the time domain
The heat in the spatial–temporal domain is calculated by
applying a numerical inverse fast Fourier transform in the
frequency domain. The computations are performed using
complex frequencies with a small imaginary part of the form
ucZuKih (with hZ0.7 Du, and Du being the frequency step)
to prevent interference from aliasing phenomena. In the time
domain, this effect is removed by rescaling the response with
an exponential window of the form eht [6]. The time variation
of the source can be arbitrary. The time Fourier transformation
of the source heat field defines the frequency domain to
be computed. The response may need to be computed from
0.0 Hz up to very high frequencies. However, as the heat
responses decay very rapidly as the frequency increases, we
may set an upper limit on the frequency for which the solution
is required.
2.2. Verification of the solution
The formulation described above was implemented and
used to compute the heat field in an unbounded medium. In
order to verify this formulation, the solution is verified in both
the frequency and time domains. In the frequency domain, this
verification is performed against solutions obtained using
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349340
a double-space Fourier transformation along the horizontal
directions (2.5D formulation). In the case of the time domain
solutions, the responses are compared with those provided by
analytical responses. When the heat source is applied at point
(X0, Y0, Z0) at time tZt0, the exact temperature evolution at
(x,y,z) is given by the expression
Tðt; x; y; zÞZeKððxKx0Þ
2CðyKy0Þ2CðzKz0Þ
2Þ=4Kt
rcð4pKtÞ3=2; if tO t0; (6)
where tZtKt0 (see Carslaw and Jaeger [4] and Banerjee [7]).
In the verification procedure, a homogeneous unbounded
medium, with thermal properties that took kZ1.4 W mK1 8CK1,
cZ880 J kgK1 8CK1 and rZ2300 kg mK3, was excited at
tZ0.0 s by a unit heat source placed at xZ0.0 m, yZ1.0 m.
The heat responses generated by a spherical unit heat source
were calculated for three different receivers: Rec. 1, Rec. 2 and
Rec. 3, placed, respectively, at (0.2,0.5,0.0) m, (0.2,0.5,0.5) m
and (0.2,0.5,1.0) m.
The calculations in this case, and in all examples
presented below for other cases, were first performed in the
frequency range [0.0, 1024.0!10K7] Hz with an increment of
DuZ1!10K7 Hz, which defines a time window of TZ2777.8 h. Complex frequencies of the form ucZuKi0.7Duhave been used to avoid the aliasing phenomenon. The spatial
period has been set as LyZ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiky=ðrcDf Þ
p. Fig. 1(a)–(c) displays
the responses in the frequency domain at the receivers Rec. 1, 2
and 3 for the first 320 frequencies: the solid lines identify the
responses using the proposed Green’s functions, while the
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
(a)
(c)
Fig. 1. Unbounded medium: frequency responses at receivers Rec. 1 (a), Rec. 2 (b) a
Temperature curves at receivers Rec. 1, Rec. 2 and Rec. 3.
marks show the response obtained using a double-space
Fourier transformation along the horizontal directions (2.5D
Green’s functions formulation). Fig. 1(d) illustrates the good
agreement between the exact time solution calculated given by
Eq. (6), represented by solid lines, and the response obtained
using the proposed Green’s functions, symbolized by marks,
for the same three receivers. The mean errors shown in
Fig. 1(d) obtained from the average of the difference between
the proposed solutions and the analytical time domain solutions
for these three receivers are, respectively, 6.66!10K12,
6.46!10K12 and 6.30!10K12 8C.
3. Green’s functions in a half-space
In this section, a semi-infinite medium bounded by a surface
with null heat fluxes or null temperatures is considered. The
required Green’s functions of the proposed formulation for a
half-space can be expressed as the sum of the surface terms
and the source terms. The surface terms need to satisfy the
boundary condition of the surface (null heat fluxes or null
temperatures), while the source terms are equal to those
presented for the infinite unbounded medium. The
surface terms for a heat source located at (x0,y0,z0) can be
expressed by
T1ðu; x; y; zÞZKip
Lyk
XNnZ1
An
knnn
J0ðknrÞeKinnjyj: (7)
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
0
0.5x10-7
1.0x10-7
1.5x10-7
2.0x10-7
2.5x10-7
0 150 300 450
Rec. 1 (0.2, 0.5, 0.0)mRec. 2 (0.2, 0.5, 0.5)mRec. 3 (0.2, 0.5, 1.0)m
Time (h)
Tem
pera
ture
(ºC
)
(b)
(d)
nd Rec. 3 (c) for a spherical unit heat source applied at point (0.0,1.0,0.0) m. (d)
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349 341
An is the unknown coefficient to be computed, so that the
heat field produced simultaneously by the source
T incðu; x; y; zÞZKip
Lyk
XNnZ1
J0ðknrÞknnn
eKinnjyKy0j
!;
and surface terms should produce T1ðu; x; y; zÞZ0 or vT1ðu;
x; y; zÞ=vyZ0 at yZ0.
The computation of the unknown coefficient is obtained for
each value of n. These coefficients are given below for the two
cases of null heat fluxes and null temperatures at the surface
yZ0.
Null normal flux at yZ0
An Z eKinny0 :
Null temperature at yZ0
An ZKeKinny0 : (8)
Replacing these coefficients in Eq. (7), we may compute the
heat terms associated with the surface.
Null normal flux at yZ0
T1ðu; x; y; zÞZKip
Lyk
XNnZ1
knnn
J0ðknrÞeKinnjyCy0j:
Null temperature at yZ0
T1ðu; x; y; zÞZip
Lyk
XNnZ1
knnn
J0ðknrÞeKinnjyCy0j: (9)
The final fundamental solutions for a half-space are given
by adding these terms, the source and the surface terms, which
leads to:
Null normal flux at yZ0
Tðu; x; y; zÞZKip
Lyk
XNnZ1
knnn
J0ðknrÞðeKinnjyj CeKinnjyCy0jÞ: (10)
Null temperature at yZ0
Tðu; x; y; zÞZKip
Lyk
XNnZ1
knnn
J0ðknrÞðeKinnjyjKeKinnjyCy0jÞ: (11)
The final heat field in the time domain is calculated by
applying a numerical inverse fast Fourier transform in the
frequency domain.
The time domain solution can also be obtained using the
image model technique. In this approach, the solution can be
achieved by superposing the heat field generated by virtual
sources with positive or negative polarity in such a way as to
simulate null temperatures or null heat fluxes at the surface.
Null normal flux at yZ0
Tðt; x; y; zÞZeðKr0=4KtÞ CeðKr1=4KtÞ
rcð4pKtÞ3=2:
Null temperature at yZ0
Tðt; x; y; zÞZeðKr0=4KtÞKeðKr1=4KtÞ
rcð4pKtÞ3=2(12)
in which tZtKt0, r0Z ðxKx0Þ2C ðyKy0Þ
2C ðzKz0Þ2 and
r1Z ðxKx0Þ2C ðyCy0Þ
2C ðzKz0Þ2.
3.1. Verification of the solution
The solutions derived before [3], using a double space
Fourier transformation along the horizontal directions (2.5D
formulation), which entails higher computational costs, are used
to verify the response in the frequency domain. The time domain
solutions are also evaluated using the image model technique.
The verification of the solution is illustrated for a
homogeneous half-space medium with thermal material proper-
ties that take kZ1.4 W mK1 8CK1, cZ880.0 J kgK1 8CK1 and
rZ230 kg mK3. This structure is excited at (0.0,1.0,0.0) m by
a point heat source. Fig. 2 gives the total amplitudes (source
and surface terms summation) in the frequency domain
obtained at the receivers Rec. 1 and Rec. 2, placed at
(0.2,0.5,0.0) m and (0.2,0.5,0.5) m, for both null normal flux
and null temperatures at yZ0, in the frequency range [0.0,
320.0!10K7] Hz. In this plot, the real and imaginary parts of
the response are given: solid lines represent the results provided
by the proposed solutions, while the marks correspond to the
2.5D formulation solutions. These results show that the
responses are similar.
In addition, the results in the time domain provided by the
proposed formulation are compared with those obtained
directly in the time domain using the image model technique,
at three different receivers (see Fig. 3). This figure shows
similar results for the image model technique solutions, which
are plotted with solid lines, and the proposed formulation
results, represented by marks. The mean errors of the results
shown in Fig. 3, obtained from the average of the difference
between the proposed solutions and the analytical time domain
solutions for these three receivers, are listed in Table 1.
4. Green’s functions in a slab formation
For a slab structure with thickness h, the Green’s functions
can be found taking into account the null heat fluxes or null
temperatures prescribed at each surface. They can be expressed
by adding the surface and source terms, which are equal to
those in the full-space.
Three scenarios can be considered: null heat fluxes at the top
and bottom interfaces (Case I); null temperatures in both
surface boundaries (Case II), and different conditions at each
surface (Case III). At the top and bottom interfaces, surface
terms can be generated and expressed in a form similar to that
of the source term.
Top surface medium
T1ðu; x; y; zÞZKip
Lyk
XNnZ1
Atn
knnn
J0ðknrÞeKinnjyj:
Bottom surface medium
T2ðu; x; y; zÞZKip
Lyk
XNnZ1
Abn
knnn
J0ðknrÞeKinnjyKhj: (13)
Atn and Ab
n are as yet unknown coefficients to be determined
by imposing the appropriate boundary conditions, so that the
field originated simultaneously by the source and the surface
0
0.5x10-7
1.0x10-7
1.5x10-7
2.0x10-7
2.5x10-7
0 150 300 450
Rec. 1 (0.2, 0.5, 0.0)mRec. 2 (0.2, 0.5, 0.5)mRec. 3 (0.2, 0.5, 1.0)m
Time (h)
Tem
pera
ture
(˚C
)
0
0.5x10-7
1.0x10-7
1.5x10-7
2.0x10-7
2.5x10-7
0 150 300 450
Rec. 1 (0.2, 0.5, 0.0)mRec. 2 (0.2, 0.5, 0.5)mRec. 3 (0.2, 0.5, 1.0)m
Time (h)
Tem
pera
ture
(˚C
)
(a) (b)
Fig. 3. Temperature curves for a half-space formation at receivers Rec. 1, Rec. 2 and Rec. 3, when a heat source is applied at point (0.0,1.0,0.0) m: (a) null normal
flux at yZ0; (b) null temperature at yZ0.
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
(a) (b)
(c) (d)
Fig. 2. Real and imaginary parts of the response for a half-space formation, when a heat source is applied at point (0.0,1.0,0.0) m: (a) Receiver 1 for null normal flux
at yZ0; (b) Receiver 2 for null normal flux at yZ0; (c) Receiver 1 for null temperature at yZ0; (d) Receiver 2 for null temperature at yZ0.
Table 1
Time domain mean errors for a half-space formation
Null normal flux at yZ0 Null temperature at yZ0
Rec. 1 1.27!10K11 8C 8.89!10K13 8C
Rec. 2 1.22!10K11 8C 4.15!10K13 8C
Rec. 3 1.25!10K11 8C 4.03!10K13 8C
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349342
terms guarantees null heat fluxes or null temperatures at yZ0
and h.
This problem formulation leads to a system of two equations
in the two unknown constants for each value of n.
Case I: null heat fluxes at the top and bottom surfaces
Kinn inn eKinnh
Kinn eKinnh inn
" #Atn
Abn
" #Z
Kinn eKinny0
inn eKinnjhKy0j
" #: (14)
Case II: null temperatures at the top and bottom surfaces
1 eKinnh
eKinnh 1
" #Atn
Abn
" #Z
KeKinny0
KeKinnjhKy0j
" #: (15)
Case III: null heat fluxes at the top surface and null
temperatures at the bottom surface
Kinn inn eKinnh
eKinnh 1
� �Atn
Abn
" #Z
Kinn eKinny0
KeKinnjhKy0j
" #: (16)
Once this system of equations has been solved, the
amplitude of the surface terms has been fully defined, and
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349 343
the heat in the slab can thus be found. The final expressions for
the Green’s functions are then derived from the sum of the
source terms and the surface terms originated in the two slab
surfaces, which leads to the following expressions
Tðu; x; y; zÞZeKi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKðiu=KÞÞ
pr0
2kr0
CKip
Lyk
XNnZ1
J0ðknrÞknnn
!ðAtn eKinnjyj CAb
n eKinnjyKhjÞ: (17)
Once again, the heat field in the time domain can be
calculated from a numerical inverse fast Fourier transform
applied in the frequency domain.
In addition, the time solution can be achieved by super-
posing the heat field generated by virtual sources with positive
or negative polarity (image model technique), and located to
ensure the desired boundary conditions. For a slab, the image
model technique needs a set of virtual sources to be placed in
such a way as to simulate null temperatures or null heat fluxes
at the surfaces.
Case I: null heat fluxes at the top and bottom surfaces
Tðt; x; y; zÞZeKr0=4Kt
rcð4pKtÞ3=2
CXNSnZ1
eKr1=4Kt CeKr2=4Kt CeKr3=4Kt CeKr4=4Kt
rcð4pKtÞ3=2:
(18)
Case II: null temperatures at the top and bottom surfaces
Tðt; x; y; zÞZeKr0=4Kt
rcð4pKtÞ3=2
CXNSnZ1
KeKr1=4KtKeKr2=4Kt CeKr3=4Kt CeKr4=4Kt
rcð4pKtÞ3=2:
(19)
Case III: null heat fluxes at the top surface and null
temperatures at the bottom surface
Tðt; x; y; zÞZeKr0=4Kt
rcð4pKtÞ3=2
CXNSnZ1
ðK1ÞnK1 eKr1=4KtKeKr2=4KtKeKr3=4KtKeKr4=4Kt
rcð4pKtÞ3=2
(20)
in which tZtKt0, r0Z ðxKx0Þ2C ðyKy0Þ
2C ðzKz0Þ2, r1Z
ðxKx0Þ2C ðyCy0C2hðnK1ÞÞ2C ðzKz0Þ
2, r2Z ðxKx0Þ2C
ðyCy0K2hnÞ2C ðzKz0Þ2, r3Z ðxKx0Þ
2C ðyKy0C2hnÞ2CðzKz0Þ
2 and r4Z ðxKx0Þ2C ðyKy0K2hnÞ2C ðzKz0Þ
2.
4.1. Verification of the solution
The formulation described above for the slab formation was
used to compute the responses at a receiver placed in a slab
2.0 m thick. The frequency domain results for the three
boundary situations assumed were then compared with those
given by a 2.5D formulation [3] for two different receivers:
Rec. 1, placed at (0.2,0.5,0.0) m, and Rec. 2, localised at
(0.2,0.5,0.5) m. An additional verification is performed in the
time domain against the solutions provided by the image model
technique described above. In this verification procedure, the
medium properties remain the same as those assumed for the
half-space (kZ1.4 W mK1 8CK1, cZ880.0 J kgK1 8CK1 and
rZ2300.0 kg mK3). The slab is heated by a harmonic point
heat source applied at (0.0,1.0,0.0) m.
Fig. 4 shows the real and imaginary parts of the total
responses, given by the sum of the source and surface terms,
at the receivers Rec. 1 and Rec. 2 for the three different
cases described above. The solid lines represent the
proposed frequency responses, while the marked points
correspond to the 2.5D formulation solution [3]. The results
confirm that the solutions to the three cases are in very close
agreement.
Fig. 5 provides the time domain solutions calculated by the
proposed formulation and identified with marks. These are in
good agreement with the results obtained with the image model
technique and are plotted using solid lines. The calculations
were obtained at three different receivers for the three cases
with their different surface conditions. The mean errors of the
results shown in Fig. 5, obtained from the average of the
difference between the proposed solutions and the analytical
time domain solutions for these three receivers, are listed in
Table 2.
5. Green’s functions in a layered formation
The solutions for more complex structures, such as a layer
over a half-space, a layer bounded by two semi-infinite media
and a multi-layer can be established imposing the required
boundary conditions at the interfaces and at the free surface.
5.1. Layer over a half-space
Assuming the presence of a layer, h1 thick, over a half-space,
we may prescribe null temperature or null heat fluxes at the free
surface (top), while at the interface we need to satisfy the
continuity of temperature and normal heat fluxes. The solution
is again expressed as the sum of the source terms (the incident
field) equal to those in the full-space and the surface terms. At
the interfaces 1 and 2, surface terms are generated, which can be
expressed in a form analogous to that of the source term.
Layer interface 1
T11ðu; x; y; zÞZKip
Lyk1
XNnZ1
Atn1
knnn1
J0ðknrÞeKinn1jyj: (21)
Layer interface 2
T12ðu; x; y; zÞZKip
Lyk1
XNnZ1
Abn1
knnn1
J0ðknrÞeKinn1jyKh1j: (22)
Half-space (interface 2)
T21ðu; x; y; zÞZKip
Lyk2
XNnZ1
Atn2
knnn2
J0ðknrÞeKinn2jyKh1j; (23)
where nnjZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKðiu=KjÞKk2
n
qwith Im(nnj)%0 and h1 is the layer
thickness (jZ1 stands for the layer (medium 1), while jZ2
Rec. 1 Rec. 2
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
(a)
(b)
(c)
Fig. 4. Real and imaginary parts of the responses for a slab formation at receivers Rec. 1 and Rec.2, when a heat source is applied at point (0.0,1.0,0.0) m: (a) Case I
(null heat flux at the top and bottom surfaces); (b) Case II (null temperature at the top and bottom surfaces); and (c) Case III (null temperature at the top surface and
null heat flux at the bottom surface).
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349344
indicates the half-space (medium 2)). Meanwhile, KjZ(kj/rjcj)
is the thermal diffusivity in the medium j (kj, rj and cj are the
thermal conductivity, the density and the specific heat of the
material in the medium j, respectively).
The coefficients Atn1, Ab
n1 and Atn2 are as yet unknown. They
are defined in order to ensure the appropriate boundary
conditions: the field produced simultaneously by the source
and surface terms allows the continuity of heat fluxes and
temperatures at yZh1, and null heat fluxes (Case I) or null
temperatures (Case II) at yZ0.
Imposing the three stated boundary conditions for each
value of n, a system of three equations in the three unknown
coefficients is defined.
Case I: null heat fluxes at yZ0.
Kinn1 inn1 eKinn1h1 0
Kinn1 eKinn1h1 inn1 inn1
eKinn1h1 1 Kk1nn1
k2nn2
266664
377775
Atn1
Abn1
Atn2
264
375Z
b1
b2
b3
24
35;
(24)
where
b1ZKnn1 eKinn1y0 ,
b2Z inn1 eKinn1jh1Ky0j,
b3ZKeKinn1jh1Ky0j, when the source is in the layer (y0!h1),
0
0.5x10-7
1.0x10-7
1.5x10-7
2.0x10-7
2.5x10-7
0 150 300 450
Rec. 1 (0.2, 0.5, 0.0)mRec. 2 (0.2, 0.5, 0.5)mRec. 3 (0.2, 0.5, 1.0)m
Time (h)
Tem
pera
ture
(˚C
)
0
0.5x10-7
1.0x10-7
1.5x10-7
2.0x10-7
2.5x10-7
0 150 300 450
Rec. 1 (0.2, 0.5, 0.0)mRec. 2 (0.2, 0.5, 0.5)mRec. 3 (0.2, 0.5, 1.0)m
Time (h)
Tem
pera
ture
(˚C
)0
0.5x10-7
1.0x10-7
1.5x10-7
2.0x10-7
2.5x10-7
0 150 300 450
Rec. 1 (0.2, 0.5, 0.0)mRec. 2 (0.2, 0.5, 0.5)mRec. 3 (0.2, 0.5, 1.0)m
Time (h)
Tem
pera
ture
(˚C
)
(a) (b)
(c)
Fig. 5. Temperature curves for a slab formation at receivers Rec. 1 and Rec.2, when a heat source is applied at the point (0.0,1.0,0.0) m: (a) Case I (null heat flux at
the top and bottom surfaces); (b) Case II (null temperature at the top and bottom surfaces); and (c) Case III (null temperature at the top surface and null heat flux at
the bottom surface).
Table 2
Time domain mean errors for a slab formation
Case I Case II Case III
Rec. 1 3.16!10K11 8C 1.00!10K12 8C 8.33!10K13 8C
Rec. 2 3.13!10K11 8C 4.30!10K13 8C 1.81!10K13 8C
Rec. 3 3.05!10K11 8C 4.24!10K13 8C 1.70!10K13 8C
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349 345
while
b1Z0,
b2Z inn1 eKinn2jh1Ky0j,
b3Z ðk1nn1=k2nn2ÞeKinn2jh1Ky0j, when the source is in the half-
space (y0Oh1).
Case II: null temperatures at yZ0.
1 eKinn1h1 0
Kinn1 eKinn1h1 inn1 inn1
eKinn1h1 1 Kk1nn1
k2nn2
266666664
377777775
Atn1
Abn1
Atn2
2664
3775Z
b1
b2
b3
264
375; (25)
where
b1ZKeKinn1y0 ,
b2Z inn1 eKinn1jh1Ky0j,
b3ZKeKinn1jh1Ky0j, when the source is in the layer (y0!h1),
while
b1Z0,
b2Z inn1 eKinn2jh1Ky0j,
b3Z ðk1nn1=k2nn2ÞeKinn2jh1Ky0j, when the source is in the half-
space (y0Oh1).
The heat field produced within the two media results from
the contribution of both the surface terms generated at the
various interfaces and the source term.
y0!h1 (source in the medium 1)
Tðu; x; y; zÞZeKi
ffiffiffiffiffiffiffiffiffiffiffiffiffiKðiu=K1Þ
pr0
2k1r0
CKip
Lyk1
XNnZ1
J0ðknrÞknnn1
!ðAtn1 eKinn1jyjCAb
n1 eKinn1jyKhjÞ; if y!h1;
Tðu; x; y; zÞZKip
Lyk2
XNnZ1
Atn2
knnn2
J0ðknrÞeKinn2jyKh1j; if yOh1;
(26)
y0Oh1 (source in the medium 2)
Tðu;x;y;zÞZKip
Lyk1
XNnZ1
J0ðknrÞknnn1
ðAtn1 eKinn1jyjCAb
n1 eKinn1jyKhjÞ;
if y!h1;
Tðu;x;y;zÞZeKi
ffiffiffiffiffiffiffiffiffiffiffiffiffiKðiu=K2Þ
pr0
2k2r0
CKip
Lyk2
!XNnZ1
Atn2
knnn2
J0ðknrÞeKinn2jyKh1j; if yOh1:
(27)
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349346
5.2. Layer bounded by two semi-infinite media
For the case of the layer placed between two semi-infinite
media, the solution also needs to account for the continuity of
temperature and heat fluxes at interface 1, since heat
propagation also occurs through the top semi-infinite space
(medium 0), which can be expressed by
T02ðu; x; y; zÞZKip
Lyk0
XNnZ1
Abn0
knnn0
J0ðknrÞeKinn0jyj; (28)
where nn0ZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKðiu=K0ÞKk2
n
pwith Im(nn0)%0.
The surface terms produced in interfaces 1 and 2 in media 1
and 2 (bottom semi-infinite medium) are expressed in Eqs.
(21)–(23).
The coefficients Abn0, At
n1, Abn1 and At
n2 are defined by
respecting the continuity of heat fluxes and temperatures at yZh1 and 0. These surface conditions lead to the following system
of four equations when the heat source is placed within the layer
K1 K1 eKinn1h1 0
Kk1nn1
k0nn0
1 eKinn1h1 0
0 KeKinn1h1 1 1
0 eKinn1h1 1 Kk1nn1
k2nn2
26666666664
37777777775
Abn0
Atn1
Abn1
Atn2
26664
37775Z
b1
b2
b3
b4
2664
3775;
(29)
where
b1ZKeKinn1y0 ;
b2ZKeKinn1y0 ;
b3ZeKinn1jh1Ky0j;
b4ZKeKinn1jh1Ky0j, when the source is in the layer (0!y0!h1),
while
b1ZeKinn0jy0j,
b2ZKðk1nn1=k0nn0ÞeKinn0jy0j,
b3Z0,
b4Z0, when the source is in the half-space (y0!h1),
and
b1Z0,
b2Z0,
b3ZeKinn2jh1Ky0j,
b4Z ðk1nn1=k2nn2ÞeKinn2jh1Ky0j, when the source is in the half-
space (y0Oh1).
The temperatures for the three media are then computed by
adding the contribution of the source terms to those associated
with the surface terms originated at the various interfaces. This
procedure produces the following expressions for the
temperatures in the three media
Tðu; x; y; zÞZKip
Lyk0
XNnZ1
Abn0
knnn0
J0ðknrÞeKinn0jyj; if y!0;
Tðu; x; y; zÞZeKi
ffiffiffiffiffiffiffiffiffiffiffiffiffiKðiu=K1Þ
pr0
2k1r0
CKip
Lyk1
XNnZ1
J0ðknrÞknnn1
!ðAtn1 eKinn1jyj CAb
n1 eKinn1jyKhjÞ; if 0!y!h1;
Tðu; x; y; zÞZKip
Lyk2
XNnZ1
Atn2
knnn2
J0ðknrÞeKinn2jyKh1j; if yOh1:
(30)
The derivation presented assumed that the heat source is
placed within the layer. However, the equations can be easily
manipulated to accommodate another position of the source.
5.3. Multi-layer
The Green’s functions for a multi-layer are established
using the required boundary conditions at all interfaces.
Consider a system built from a set of m plane layers of
infinite extent bounded by two flat, semi-infinite media, as
shown in Fig. 6. The top semi-infinite medium is called
medium 0, while the bottom semi-infinite medium is assumed
to be the medium mC1. The thermal material properties and
thickness of the various layers may differ. The system of
equations is achieved considering that the multi-layer is excited
by a heat source located in the first layer (medium 1). The heat
field is computed at some position in the domain, taking into
account both the surface heat terms generated at each interface
and the contribution of the heat source term.
For the layer j, the heat surface terms on the upper and lower
interfaces can be expressed as
T j1ðu; x; y; zÞZKip
Lykj
XNnZ1
Atnj
knnnj
J0ðknrÞeKinnjjyK
PjK1
lZ1
hlj
;
T j2ðu; x; y; zÞZKip
Lykj
XNnZ1
Abnj
knnnj
J0ðknrÞeKinnjjyK
PjlZ1
hlj
;
(31)
where nnjZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKiu=KjÞKk2
n
q, with Im(nnj)%0 and h1 is the
thickness of the layer l. The heat surface terms produced at
interfaces 1 and mC1, which govern the heat that propagates
through the top and bottom semi-infinite media, are,
respectively, expressed by
T02ðu; x; y; zÞZKip
Lyk0
XNnZ1
Abn0
knnn0
J0ðknrÞeKinn0jyj; and
T ðmC1Þ2ðu; x; y; zÞZKip
LykðmC1Þ
!XNnZ1
AtnðmC1Þ
knnnðmC1Þ
J0ðknrÞeKinnðmC1ÞjyK
PmlZ1
hlj
:
(32)
A system of 2(mC1) equations is derived, ensuring the
continuity of temperatures and heat fluxes along the mC1
Fig. 6. Geometry of the problem for a multi-layer bounded by two semi-infinite
media.
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349 347
interfaces between layers. Each equation takes into account the
contribution of the surface terms and the involvement of the
incident field. All the terms are organized according to the form
FaZb
1 1 KeKinn1h1 / 0 0 0
1
k0nn0
K1
k1nn1
KeKinn1h1
k1nn1
/ 0 0 0
0 KeKinn1h1 1 / 0 0 0
0eKinn1h1
k1nn1
1
k1nn1
/ 0 0 0
/ / / / / / /
0 0 0 / 1 KeKinnmhm 0
0 0 0 / K1
kmnnmK
eKinnmhm
kmnnm0
0 0 0 / KeKinnmhm 1 1
0 0 0 /eKinnmhm
kmnnm
1
kmnnmK
1
kðmC1ÞnnðmC1Þ
2666666666666666666666666666666664
3777777777777777777777777777777775
Abn0
Atn1
Abn1
/
Atnm
Abnm
AtnðmC1Þ
2666666666666666664
3777777777777777775
Z
eKinn1y0
1
k1nn1
eKinn1y0
eKinn1 jh1Ky0 j
K1
k1nn1
eKinn1jh1Ky0 j
/
0
0
0
0
266666666666666666666666666666666664
377777777777777777777777777777777775
:
(33)
The resolution of the system gives the amplitude of the
surface terms in each interface. The temperature field for each
layer formation is obtained by adding these surface terms to the
contribution of the incident field, leading to the following
equations.
Top semi-infinite medium (medium 0)
Tðu; x; y; zÞZKip
Lyk0
XNnZ1
Abn0
knnn0
J0ðknrÞeKinn0jyj; if y!0;
layer 1 (source position)
Tðu; x; y; zÞZeKi
ffiffiffiffiffiffiffiffiffiffiffiffiffiKðiu=K1Þ
pr0
2k1r0
CKip
Lyk1
XNnZ1
J0ðknrÞknnn1
!ðAtn1 eKinn1jyjCAb
n1 eKinn1jyKh1jÞ; if 0!y!h1;
layer j (js1)
Tðu; x; y; zÞZKip
Lykj
XNnZ1
J0ðknrÞ
!knnnj
Atnj e
KinnjjyKPjK1
lZ1
hlj
CAbnj e
KinnjjyKPjlZ1
hlj
0@
1A;
ifXjK1
lZ1
hl!y!XjlZ1
hl;
bottom semi-infinite medium (medium mC1)
Tðu; x; y; zÞZKip
LykðmC1Þ
!XNnZ1
AtnðmC1Þ
knnnðmC1Þ
J0ðknrÞeKinnðmC1ÞjyK
PmlZ1
hlj
:
(34)
Notice that when the position of the heat source is changed,
the matrix F remains the same, while the independent terms of
b are different. However, as the equations can be easily
manipulated to consider another position for the source, they
are not included here.
5.4. Verification of the solution
Next, the results are found for the three scenarios. First, a
flat layer, 2.0 m thick, is assumed to be bounded by one half-
space. Null heat fluxes or null temperatures are prescribed at
the top surface. Then, a flat layer, also 2.0 m thick, bounded by
two semi-infinite media, is used to evaluate the accuracy of the
proposed formulation. The thermal material properties used in
the intermediate layer are kZ1.4 W mK1 8CK1, cZ880.0 J kgK1 8CK1 and rZ2300.0 kg mK3, while at the top
and bottom media they are kZ63.9 W mK1 8CK1, cZ434.0 J kgK1 8CK1 and rZ7832.0 kg mK3.
The calculations have been performed in the frequency
domain from 0.0 to 320.0!10K7 Hz. The amplitude of the
response for four receivers placed in two different media was
computed for a heat point source applied at (0.0,1.0,0.0) m.
The real and imaginary parts of the full response at receivers
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.010
-0.005
0
0.005
0.010
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.010
-0.005
0
0.005
0.010
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
(a) (b)
(c) (d)
Fig. 7. Null heat flux at yZ0 (Case I): real and imaginary parts of the responses for a layer over a half-space at different receivers: (a) Rec. 1; (b) Rec. 2; (c) Rec. 4;
(d) Rec. 5.
-0.2
0
0.2
0.4
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.010
-0.005
0
0.005
0.010
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.010
-0.005
0
0.005
0.010
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
(a) (b)
(c) (d)
Fig. 8. Null temperature at yZ0 (Case II): real and imaginary parts of the responses for a layer over a half-space at different receivers: (a) Rec. 1; (b) Rec. 2;
(c) Rec. 4; (d) Rec. 5.
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349348
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.2
0
0.2
0.4
0.6
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.010
-0.005
0
0.005
0.010
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
-0.010
-0.005
0
0.005
0.010
0 1x10-5 2x10-5 3x10-5
Real partImaginary part
Frequency (Hz)
Am
plitu
de
(a) (b)
(c) (d)
Fig. 9. Real and imaginary parts of the responses for a layer bounded by two semi-infinite media at different receivers: (a) Rec. 1; (b) Rec. 2; (c) Rec. 4; (d) Rec. 5.
A. Tadeu, N. Simoes / Engineering Analysis with Boundary Elements 30 (2006) 338–349 349
Rec. 1 (0.2,0.5,0.0) m, Rec. 2(0.2,0.5,0.5) m, Rec. 4
(0.2,2.5,0.0) m and Rec. 5 (0.2,2.5,0.5) m are displayed in
Figs. 7–9. The solid lines represent the analytical responses
computed using the proposed Green’s functions, while the
marked points correspond to the 2.5D formulation [3] solution.
As can be seen, these two solutions seem to be in very close
agreement, and equally good results were obtained from tests
in which heat sources and receivers were situated at different
points.
6. Conclusions
Three-dimensional Green’s functions for computing the
transient heat transfer by conduction in an unbounded medium,
half-space, slab and layered media have been presented. The
transient heat responses generated by a spherical heat source
were first obtained in the frequency domain. Time solutions
were then obtained after the application of an inverse Fourier
transform in the frequency domain.
The proposed analytical solutions were obtained after
writing the response of a spherical heat source as a sum of
Bessel integrals, assuming a set of sources equally spaced
along the vertical direction. The results for a layered formation
are obtained by adding the heat source term and the surface
terms required to satisfy the interface boundary conditions
(temperature and heat flux continuity).
The unbounded medium formulation was verified by
comparing its time responses with the exact known time
solution. In turn, the proposed analytical solutions used in the
half-space, slab systems and layered media formulation were
compared against the frequency responses obtained when a
double space Fourier transformation was applied along the
horizontal directions [3]. Therefore, the present formulation
involves less computational effort than the previous solution. In
addition, time responses provided by the proposed formulation
were compared with those provided by analytical solutions,
known for simpler cases such as the half-space and the slab
formations.
The proposed Green’s functions can be used as fundamental
solutions in BEM or MFS algorithms for solving layered
formations containing buried three-dimensional inclusions,
while the analytical solutions obtained before [3] are better
suited to resolving the case of layered formulations containing
cylindrical inclusions.
References
[1] Sommerfeld A. Mechanics of deformable bodies. New York: Academic
Press; 1950.
[2] Ewing WM, Jardetzky WS, Press F. Elastic waves in layered media. New
York: McGraw-Hill; 1957.
[3] Tadeu A, Antonio J, Simoes N. 2.5D Green’s functions in the frequency
domain for heat conduction problems in unbounded, half-space, slab and
layered media. CMES: Comput Model Eng Sci 2004;6(1):43–58.
[4] Carslaw HS, Jaeger JC. Conduction of heat in solids. 2nd ed. New York:
Oxford University Press; 1959.
[5] Bouchon M, Aki K. Discrete wave-number representation of seismic-
source wave field. Bull Seismol Soc Am 1977;67:259–77.
[6] Kausel E, Roesset JM. Frequency domain analysis of undamped systems.
J Eng Mech ASCE 1992;118(4):721–34.
[7] Banerjee PK. The boundary element methods in engineering. England:
McGraw-Hill; 1981.